I voted 3 and if you bear with me, here is the reason why.
One piece rods are too long too fit in my car and expect passengers to be comfortable...especially rigged.
Two piece rods are great for breaking down while rigged and moving from one spot to another. Try doing that with a 3 or 5 piece. I've heard of rods missing sections that disappeared while trekking unrigged. Not a good feeling.
Three piece rods, especially vintage, often have a smoother bend and improved casting.
But the disadvantage is as has already been noted.
Four piece rods are great for packing, AND can be broken down conveniently as a two piece.
Five piece rods are similar to three piece in action, but can be broken down to be even more packable. Again, the same disadvantage as a 3 piece.
Six piece rods are great for packing in a suitcase and can be broken down while rigged, similar to two piece.
Other than Tenkara, I don't have any fly rods with more pieces than six.
So anyway, I was reflecting on my decision and I accidently hit 3. Oh well.
If I hadn't, it would been 4 piece.